Description
Abstract: The exponential representation of the S-matrix behaves in a relatively simple way under the classical limit, allowing a classical function on phase space we call the Magnusian’’ to appear and act as a generating function of all classical observables, thus defining aclassical S-matrix’’ through the exponentiated action of the Poisson bracket. The Magnusian is naturally an in-in quantity, distinct from but related to the in-out on-shell action, and is a generalization of a notion of the eikonal that has been gaining popularity in amplitudes literature. It is naturally computed by a diagrammatic ``Magnus series’’ with well-understood iepsilons. This diagrammatic expansion naturally pre-incorporates all physical implications of unitarity, so that perturbative results at a fixed order are exactly unitary, rather than only being so up to higher order corrections. The Magnusian can be defined even outside of a scattering context, and in a general setting it provides the canonically-compatible way to time-average degrees of freedom within the Hamiltonian.